It was then appropriate, with this electronic reedition of the monograph, to have three appendices which could illustrate how that fundamental inversion formula was implemented in other environments, explicitly and also implicitly.
In the first appendix ("Inversions de Möbius") it is shown how to go from the Möbius inversion formula for a partially commutative monoid to the Möbius formula for a locally finite partially ordered set, and conversely.
In the second appendix Bodo Lass shows that by means of a simple specialization of the variables the fundamental inversion formula provides a noncommutative version of the celebrated chromatic polynomial identity for graphs: (-1)|V|\chiG(-1)=a(G).
The third appendix, written by Christian Krattenthaler, presents Viennot's theory of heaps of pieces, a theory that has been very fruitful in the combinatorial theory of orthogonal polynomials and in the calculation of multivariable generating functions for polyominoes. The equivalence of the theory of heaps and the theory of partially commutative monoids is explicitly established.
Dominique Foata, July 2006
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